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2003 Stability of standing waves for nonlinear Schrödinger equations with potentials
Reika Fukuizumi, Masahito Ohta
Differential Integral Equations 16(1): 111-128 (2003). DOI: 10.57262/die/1356060699


We study the stability of standing waves $e^{i \omega t}\phi_{\omega}(x)$ for a nonlinear Schrödinger equation with an attractive power nonlinearity $|u|^{p-1}u$ and a potential $V(x)$ in $\mathbb R^n$. Here, $\omega\in \mathbb R$ and $\phi_{\omega}(x)$ is a ground state of the stationary problem. Under suitable assumptions on $V(x)$, we show that $e^{i \omega t}\phi_{\omega}(x)$ is stable for $p <1+4/n$ and sufficiently large $\omega$, or for $1 <p <2^*-1$ and $\omega$ close to $-\lambda_1$, where $\lambda_1$ is the lowest eigenvalue of the operator $-\Delta+V(x)$. We give an improvement of previous results such as Rose and Weinstein [19], or Grillakis, Shatah and Strauss [11], for unbounded potentials $V(x)$ which cannot be treated by the standard perturbation argument.


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Reika Fukuizumi. Masahito Ohta. "Stability of standing waves for nonlinear Schrödinger equations with potentials." Differential Integral Equations 16 (1) 111 - 128, 2003.


Published: 2003
First available in Project Euclid: 21 December 2012

zbMATH: 1031.35132
MathSciNet: MR1948875
Digital Object Identifier: 10.57262/die/1356060699

Primary: 35Q55
Secondary: 35A15 , 35B35

Rights: Copyright © 2003 Khayyam Publishing, Inc.

Vol.16 • No. 1 • 2003
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